Răspuns:
3645
Explicație pas cu pas:
a,b,c,d cifre in baza zece
[tex]\overline {abcd} = {a}^{6} \cdot d[/tex]
[tex]1 \leqslant a \leqslant 9 \iff {1}^{6} \leqslant {a}^{6} \leqslant {9}^{6} \\ 1 \leqslant d \leqslant 9[/tex]
[tex]{5}^{6} \cdot 1 > 9999 \implies {a}^{6} \leqslant {4}^{6} \\ {2}^{6} \cdot 9 < 1000 \implies {3}^{6} \leqslant {a}^{6}[/tex]
=> a ∈ {3; 4}
[tex]a = 3 \iff \overline {3bcd} = {3}^{6} \cdot d \\ \overline {3bcd} = 729 \cdot d \\ 3000 < \overline {3bcd} < 3999 \\ \iff 3000 < 729 \cdot d < 3999 \\d = 5 \implies 729 \cdot 5 = 3645 \\ [/tex]
[tex]a = 4 \iff \overline {4bcd} = {4}^{6} \cdot d \\ \overline {4bcd} = 4096 \cdot d \\ 4000 < \overline {4bcd} < 4999 \\ \iff 4000 < 4096 \cdot d < 4999 \\ d = 1 \implies 4096 \cdot 1 = 4096, 6 \not = 1[/tex]
=> unica soluție este:
[tex]\overline {abcd} = 3645[/tex]
